## solvers in problem decomposition of the modern financial mathematics (notes from lectures)

Kronecker product – replace a(ij) by a block a(ij)B, for matrices A and B

Matrices from partial diff equations are sparse

Forward Euler initial value point is a stable method

First-order initial value methods are not accurate

Trapezoidal rule, Adams-Bashforth or backward diffs, ie. second-order ones, may be unstable

Generalized backward differences, both explicit and implicit,

Consider flash of light and water block problems for wave equation

Four methods to test difference approximations: forward, centered, Lax-Friedrichs, Lax-Wendroff

Use of staggered grids to solve diffs equations- why not other meshing with dimension reduction?

Growth equation based on the wave function, sinc squered, semidiscrete growth

Local discretization errors and CFL conditions of the Leapfrog method

KKT primal-dual and Lagrange multipliers for solving auction theory problems

Infinite signal speed, energy dissipation and heat conservation tackled with the heat equation

Definite boundary definition, e.g.insulated or aborbing boundaries

Convection-diffusion in three modes: centered explicit, upwind convection, implicit diffiusion

Fast marching method for meshpoint or lattice distance calc

Lagrangian follows particles, Eulerian in promiscuous mode

Elimination in sparse matrices: fill in, Kronecker product, red-black permutation, minimum degree algo

Iterative Jacobi: slow convergence for low frequencies

Gauss-Seidel iteration (anyone know poker player Eric Seidel? any family?), V-cycle vs full multigrid cost

Residual restriction may result in aliasing

Multigrid reduces low frequencies